I'm totally lost on this problem, which is from a set of notes by Tom Leinster on Category Theory:
Let $\phi: G \to H$ be a homomorphism of groups. Associated with $\phi$ are the diagrams $ker(\phi) \xrightarrow{i} G \xrightarrow{\phi} H$ and $ker(\phi) \xrightarrow{i} G \xrightarrow{\epsilon} H$ where $i$ is the inclusion map and $\epsilon$ is the trivial homomorphism.
It then says that the map $i$ satisfies $\phi \circ i = \epsilon \circ i$ and is 'universal as such'. The exercise is to make this precise.
One way to make this statement precise is to show that $i: \ker \phi \rightarrow G$ is final in a specific category.
Consider the category whose objects are pairs $(f,A)$ where $f \in \operatorname{Hom}_{Grp}(A,G)$ and $\phi \circ f$ trivial. Morphisms between objects $(f,A)$ and $(g,B)$ are commutative diagrams
$$\require{AMScd} \begin{CD} A @>{f}>> G\\ @VV\sigma V @V \mathbb 1 VV \\ B @>{g}>> G \end{CD}$$
where $\sigma$ is a group homomorphism. You can prove that $(i, \ker \phi)$ is a final object of this category that we have cooked up.
A little thought shows that Zach L's answer agrees with this characterization. The statement that every morphism $f: H \rightarrow G$ with $\phi \circ f = \epsilon \circ f$ factors uniquely via a morphism $F: H \rightarrow \ker \phi$ such that $ i \circ F =f$ is equivalent to the statement that $(i,\ker \phi)$ is final.
This might be unduly complex for dealing with kernels alone -- but, for me, approaching universal properties in this way is helpful. I've been reading Aluffi's Algebra book, where he introduces and utilizes very basic category theory. He associates universal properties with terminal objects of canonically constructed auxiliary categories; this serves to clarify the somewhat amorphous (in my experience) notion of a universal property while alluding to appropriate generalizations of familiar concepts (like that of a kernel) to other categories.